Taubes nonvanishing theorem

A closed symplectic manifold of real dimension four with $b_2^{+}>1$ has a nonzero Seiberg–Witten invariant of a four-manifold in its canonical structure, with value $\pm 1$. The positive index of the intersection form hypothesis avoids chamber dependence. This provides the symplectic connected-sum obstruction.

A closed symplectic manifold of real dimension four cannot be a connected sum of two oriented smooth four-manifolds each with positive positive index of the intersection form. The total $b_2^{+}$ exceeds one; Taubes nonvanishing theorem and connected-sum vanishing of the Seiberg–Witten invariant would force the same invariant to be both nonzero and zero.